An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands
Shane Chern, James A. Sellers
TL;DR
This work establishes an infinite family of 2-adic internal congruences for the partition function PDO(n), counting odd-part partitions with designated summands, by proving $PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}$ for all $k\ge 0$ and $n\ge 0$. The authors develop a rich modular framework using generating function dissections, a degree-two unitizing operator, and a Hauptmodul $\xi$ on $X_0(6)$, deriving explicit modular equations and recurrences among auxiliary functions $\gamma$, $\delta$, $\kappa$, and $\Lambda$, all expressed in $\mathbb{Z}[\xi]$. A novel sequence of auxiliary functions $\Phi_k$ is introduced, with controlled $\xi$-powers and 2-adic growth, enabling a rigorous induction that yields the claimed divisibilities. The results extend known cases of internal congruences and reveal uniquely structured 2-adic dissection slices, contributing to the interface of partition arithmetic and modular form techniques. As a consequence, the paper also yields Ramanujan-type congruences for PDO modulo powers of 2, underscoring the practical significance of the approach for understanding divisibility properties of restricted partitions.
Abstract
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size $n$ by the function $PDO(n)$. Since then, numerous authors have proven a variety of divisibility properties satisfied by $PDO(n)$. Recently, the second author proved the following internal congruences satisfied by $PDO(n)$: For all $n\geq 0$, \begin{align*} PDO(4n) &\equiv PDO(n) \pmod{4},\\ PDO(16n) &\equiv PDO(4n) \pmod{8}. \end{align*} In this work, we significantly extend these internal congruence results by proving the following new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$, $$PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}.$$ We utilize several classical tools to prove this family, including generating function dissections via the unitizing operator of degree two, various modular relations and recurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, and an induction argument which provides the final step in proving the necessary divisibilities. It is notable that the construction of each $2$-dissection slice of our generating function bears an entirely different nature to those studied in the past literature.